# Where does 1/sqrt(2) come from in the state of i

I’m trying to learn about calculating coordinates for $$\theta$$ and $$\varphi$$ in a Bloch-sphere.

I came accross this book about it, including example questions.

At question 2.12b, they ask to give the value for theta and psi for the state |i⟩. In the answers they rewrite the formula down below.

Can anyone explain where the $$\frac{1}{\sqrt{2}}$$ comes from?

• The book has a section about all states being of unit length. I suggest to read it one more time but carefully. $1/\sqrt2$ makes sure the state is on the Bloch sphere and not outside or inside of it. Dec 30, 2022 at 20:17

The coefficients come from normalization condition. It ensures that sum of probabilities of measuring basis states forming a quantum state is equal to one. The coefficients are so-called probability amplitudes, not probabilities themselves. It holds that probability of measuring $$i$$th basis state is $$|a_i|^2$$, where $$a_i$$ is probability amplitude of the basis state. As mentioned above, it must hold that $$\sum_i |a_i|^2 = 1$$.
In your case, you have two basis states, namely $$|0\rangle$$ and $$|1\rangle$$. Probability of measuring each of them is 50% as $$1/\sqrt{2}$$ squared is $$1/2$$.